Is working at a bank Islamically permissible?

Narrated Jabir (RA):
Allah’s Messenger (ﷺ) cursed the one who accepts usury, the one who gives it, the one who records it and the two witnesses to it, saying, “They are all the same.” [Reported by Muslim].

Working at a bank will definitely revolve around one of the categories mentioned in the Hadith and makes the nature of the job Haram.
Please try your best to find a different type of job and insha’Allah Allah swt will open the doors for you.

See below the declaration of the AMJA council ( a recognized and reputable Islamic council of scholars in America ) on many job opportunities and their Islamic permissibility.

http://www.amjaonline.org/en/declarations/20-declarations/53-decisions-and-recommendations-of-amja-s-fifth-annual-convention

VII: Working at interest-based financial institutions
The basic principle on working for interest-based financial institutions is that it is unlawful, due to the curse of the Prophet (may Allah bless him and grant him peace) on the one who devours interest, the one who pays it, the one who writes [the contract] and both witnesses, and his saying, “They are all alike.” However, exigent needs are to be considered, as long as they are judged according to their extent and steps are taken to bring the condition to an end.
Fiqh councils have granted a license for anyone who does not find lawful employment to work in places where lawful and prohibited are mixed, as long as one does not deal with prohibited acts directly and that one exerts his every effort to search for another job free of doubtful matters. AMJA does not see anything to prevent this ruling from being applied to employment in interest-based financial institutions. The exemption is therefore granted to work in fields that are not directly related to interest, in writing the contracts, witnessing them, or providing any direct or intentional assistance for any of that.